Caption: Figure 11: Time history of horizontal interaction force varying maximum of difference function of dynamic stiffness coefficient
and wave amplitude attenuation.
The coefficient of support compression stiffness [k.sub.11] and the support ratcheting stiffness coefficient
[k.sub.21] would be found  using formula (20), where the coefficients [r.sub.11], [s.sub.11] are:
where [M.sub.Tr], [[alpha].sub.m] and q--the first harmonic of the moment variations at the output shaft of the gear with a ratio coefficient [k.sub.G], of the rotary angle variation of the motor shaft and of the rotary angle variation of the manipulator link respectively; [C.sub.Eq]--an equivalent stiffness coefficient
of the gear, reduced to its output shaft; [T.sub.E]--time constant of the elastic gear; p--differential operator.
Among them, [m.sub.2] is body quality, [m.sub.1] is wheel quality, [k.sub.2] is suspension stiffness, [c.sub.2] is damping coefficient, [k.sub.1] is tire equivalent stiffness coefficient
, [z.sub.2] is the displacement of body, [z.sub.1] is displacement of wheels, [z.sub.0] is the ground displacement excitation, and [f.sub.a](t - [tau]) is active control.
where the stiffness coefficient
[c.sub.s] is assumed to be uniform along the immersed boundary length.
Huang  has employed a conjugate gradient method (CGM) to solve the nonlinear inverse vibration problem for the estimation of the time-dependent stiffness coefficient
. Recently, Liu [13, 14] has developed a Lie-group shooting method to study the inverse vibration problem for estimating the time-dependent damping and stiffness coefficients
and simultaneously derived a closed-form solution to estimate the parameters.
The spring stiffness coefficient
is decided on by 
In this section, the Green's function of the uniformly distributed inclined load acting in the TI medium is derived, including both the displacement and stress, which serve the elementary solution of IBEM to obtain the stiffness coefficient
of the foundation.
In analysis model of structural pounding as shown in Figure 1, [m.sub.1] and [m.sub.2] are the qualities of two colliding spheres; [s.sub.1] and [s.sub.2] stand for the displacements of two colliding spheres; [v.sub.1] and [v.sub.2] express respective velocities of two colliding spheres at the initial moment of contact; [v'.sub.1] and [v'.sub.2] are the velocities of colliding spheres after impact; c stands for the damping coefficient; [g.sub.p] is the initial clearance of the colliding spheres; k expresses the stiffness coefficient
of impact contact element.
The VIB uses the Cauchy-Born rule of crystal elasticity to obtain macroscopic collective behaviour of the stochastic bond network inside solid material, and it only considers the stretch energy of virtual internal bond and has only one type of bond possessing only one kind of stiffness coefficient
(stretch stiffness), which lead to free rotation of bonds.
The only concern in this stage is then confidently classifying changed stiffness coefficient
. The system identification methods generally can only be applied on a set of data with limited time window.
Several factors might be responsible for observed discrepancies: the pressure sensor and the muscle are connected with 1m long line meaning that the sensor reading has to be corrected; the equation 6 for volume flow might better approximate discharge cycle than a filling cycle; the differential equation 11 makes assumption of isothermal process with ideal gas and it might be needed to refine it; the stiffness coefficient
of parallel spring was considered constant.